# How do I find a 95% percent ML confidence interval of the probability of category 5 strength

Let ${\bf X} = (X_1, \dots, X_n)$ denote the log-intensities of all $n = 291$ tropical cyclones from the north Atlantic basin between 1980 and 2006. Log-intensity is defined as the logarithm of its maximum sustained windspeed (measured in knots). A tropical cyclone is a category 5 hurricane if its maximum sustained windspeed is 137 knots or more, i.e. if its log-intensity is 4.92 or more. We are interested in the probability $p \in [0,1]$ of a tropical cyclone attaining category 5 strength. Consider the following model: $X_i \overset{{\rm iid}}{\sim} \mathsf{Normal}(\mu, \sigma^2)$, where $\sigma = 0.33, \mu \in (-\infty, \infty)$. Under this model $p = \Phi\left({\mu - 4.92 \over \sigma}\right)$ where $\Phi(x)$ denotes the standard normal CDF. Find the 95% ML confidence interval for $p$ based on the following summaries of the data $$\bar x = 4.21 \quad \sqrt{n} = 17.06 \quad \#\{x_i \ge 4.92\} = 9$$

My Work

The formula for an ML interval for $X_i \overset{{\rm iid}}{\sim} \mathsf{Normal}(\mu, \sigma^2)$ with known $\sigma$ is $B_c(x) = [\bar X \mp c{\sigma \over \sqrt{n}}]$. Calculating along these lines, for $c = 1.96$ as the $z$ value that leaves $0.05$ probability in the two tails combined of the normal pdf, I get $B_{1.96}(x) = [4.172, 4.247]$, but when I try to evaluate the endpoints of this interval for $p$ given that $\Phi(x)$ is a monotone transformation, I get nonsense answers. What am I doing wrong?

Also, this will never be handed in for any class, I simply came across this problem and wondered how it could be solved.

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So I should discard my ML interval altogether? I am confused about how a confidence interval for $p$ (the point of the question) helps me at all given that I should just take $1 - \Phi(4.92)$ according to your last suggestion –  Zvpunry Sep 29 '12 at 15:51